Currency Carry Trade with Futures





Kerry Back

Speculating on currencies

Currency speculation with futures

  • Consider selling currency futures that are in contango
  • If forward curve doesn’t move, forward roll will produce profits
  • Likewise, buying currency futures in backwardation may be profitable
  • What may go wrong?
    • Spot may rise for contango futures
    • Spot may fall for backwardation futures

Currency carry trade

  • Borrow in low interest rate currencies
  • Invest in high interest rate currencies
  • What may go wrong?
    • Low interest-rate currencies may appreciate
    • High interest-rate currencies may depreciate

Equivalence

  • Low interest-rate \(\Leftrightarrow\) contango. Appreciate \(\Leftrightarrow\) spot rises
  • High interest-rate \(\Leftrightarrow\) backwardation. Depreciate \(\Leftrightarrow\) spot falls

What do the data show?

  • Currency carry trade has been profitable on average
  • Equivalently, selling contango futures and buying backwardation futures has been profitable on average

Futures options

  • A call option on a futures contract gives you the right to acquire a long position in the futures (buy the futures) at the option strike.

  • A put option gives you the right to acquire a short position in the futures (sell the futures) at the option strike.

  • If you’ve sold an option and it is exercised, you are rolled into the opposite futures position.

    • short futures if you sold a call
    • long futures if you sold a put

  • On the exercise date of a call, daily settlement produces a cash flow of futures_settlement_price - strike to the party who exercised and the opposite cash flow to the party who sold the call.

  • For a put, it is strike - futures_settlement_price.

  • Everyone who wants could unwind the futures position by making an offsetting trade on the day of exercise.

Risk-neutral probability from before

  • If there were no risk premium, the call value would be the expected value discounted at the risk-free rate:

\[C = \frac{p \times \text{\$}5 + (1-p)\times \text{\$}0}{1.05}\]

  • where \(p=\) prob of positive return.

  • The stock price would also be the discounted expected value:

\[\text{\$}100 = \frac{p \times \text{\$}110 + (1-p) \times \text{\$}90}{1.05}\]

    • Solve the stock equation for \(p\) and substitute into the call option equation.

Risk-neutral probability for futures options

  • The futures price would be the expected (not discounted, because no cash changes hands upon purchase/sale):

\[\text{\$}100 = p \times \text{\$}110 + (1-p) \times \text{\$}90\]

Calibrating binomial tree for futures option

  • Set \(T=\) time to maturity in years of a futures option
  • Set \(N=\) number of periods in a binomial model
  • Set \(\Delta t = T/N=\) period length
  • Set the up rate of return as \(e^{\sigma\sqrt{\Delta t}}-1\) and down rate of return as \(e^{-\sigma\sqrt{\Delta t}} - 1\).

  • The risk-neutral probability of “up” is

\[\frac{1 - e^{-\sigma \Delta t}}{e^{\sigma \Delta t} - e^{-\sigma \Delta t}}\]